A pair of sufficient statistics for a normal

Question

You have independent observations X1, \dots, Xn from a normal distribution with both the mean \mu and the variance \sigma^2 unknown. Find a pair of statistics that are jointly sufficient for (\mu, \sigma^2).

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Two unknowns, expect a two-number summary. For a two-parameter model you generally cannot compress to a single number. The factorization theorem still applies: write the joint density as a parameter-piece (depending on the data only through some summary, now possibly a pair) times a data-only piece. Expand the exponent. The joint density is proportional to (\sigma^2)^{-n/2}\exp\!\Big(-\frac{1}{2\sigma^2}\sumi (xi - \mu)^2\Big). Expand the square inside: \sumi (xi - \mu)^2 = \sumi xi^2 - 2\mu\sumi xi + n\mu^2. Substituting, f(x; \mu, \sigma^2) \propto (\sigma^2)^{-n/2}\exp\!\Big(-\frac{1}{2\si…

A pair of sufficient statistics for a normal

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